Quantifying minimal noncollinearity among random points

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© 2018 Society for Industrial and Applied Mathematics and by SIAM. Let φn,K denote the largest angle in all the triangles with vertices among the n points selected at random in a compact convex subset K of ℝd with nonempty interior, where d ≥ 2. It is shown that the distribution of the random variable (formula presented),where λd (K) is a certain positive real number which depends only on the dimension d and the shape of K, converges to the standard exponential distribution as n → ∞. By using the Steiner symmetrization, it is also shown that λd(K), which is referred to in the paper as the elongation of K, attains its minimum if and only if K is a ball B(d) in Rd. Finally, the asymptotics of λd (B(d)) for large d is determined.

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Theory of Probability and its Applications